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Theorem axrep5 4767
Description: Axiom of Replacement (similar to Axiom Rep of [BellMachover] p. 463). The antecedent tells us 𝜑 is analogous to a "function" from 𝑥 to 𝑦 (although it is not really a function since it is a wff and not a class). In the consequent we postulate the existence of a set 𝑧 that corresponds to the "image" of 𝜑 restricted to some other set 𝑤. The hypothesis says 𝑧 must not be free in 𝜑. (Contributed by NM, 26-Nov-1995.) (Revised by Mario Carneiro, 14-Oct-2016.)
Hypothesis
Ref Expression
axrep5.1 𝑧𝜑
Assertion
Ref Expression
axrep5 (∀𝑥(𝑥𝑤 → ∃𝑧𝑦(𝜑𝑦 = 𝑧)) → ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤𝜑)))
Distinct variable group:   𝑥,𝑦,𝑧,𝑤
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem axrep5
StepHypRef Expression
1 19.37v 1908 . . . . 5 (∃𝑧(𝑥𝑤 → ∀𝑦(𝜑𝑦 = 𝑧)) ↔ (𝑥𝑤 → ∃𝑧𝑦(𝜑𝑦 = 𝑧)))
2 impexp 462 . . . . . . . 8 (((𝑥𝑤𝜑) → 𝑦 = 𝑧) ↔ (𝑥𝑤 → (𝜑𝑦 = 𝑧)))
32albii 1745 . . . . . . 7 (∀𝑦((𝑥𝑤𝜑) → 𝑦 = 𝑧) ↔ ∀𝑦(𝑥𝑤 → (𝜑𝑦 = 𝑧)))
4 19.21v 1866 . . . . . . 7 (∀𝑦(𝑥𝑤 → (𝜑𝑦 = 𝑧)) ↔ (𝑥𝑤 → ∀𝑦(𝜑𝑦 = 𝑧)))
53, 4bitr2i 265 . . . . . 6 ((𝑥𝑤 → ∀𝑦(𝜑𝑦 = 𝑧)) ↔ ∀𝑦((𝑥𝑤𝜑) → 𝑦 = 𝑧))
65exbii 1772 . . . . 5 (∃𝑧(𝑥𝑤 → ∀𝑦(𝜑𝑦 = 𝑧)) ↔ ∃𝑧𝑦((𝑥𝑤𝜑) → 𝑦 = 𝑧))
71, 6bitr3i 266 . . . 4 ((𝑥𝑤 → ∃𝑧𝑦(𝜑𝑦 = 𝑧)) ↔ ∃𝑧𝑦((𝑥𝑤𝜑) → 𝑦 = 𝑧))
87albii 1745 . . 3 (∀𝑥(𝑥𝑤 → ∃𝑧𝑦(𝜑𝑦 = 𝑧)) ↔ ∀𝑥𝑧𝑦((𝑥𝑤𝜑) → 𝑦 = 𝑧))
9 nfv 1841 . . . . 5 𝑧 𝑥𝑤
10 axrep5.1 . . . . 5 𝑧𝜑
119, 10nfan 1826 . . . 4 𝑧(𝑥𝑤𝜑)
1211axrep4 4766 . . 3 (∀𝑥𝑧𝑦((𝑥𝑤𝜑) → 𝑦 = 𝑧) → ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤 ∧ (𝑥𝑤𝜑))))
138, 12sylbi 207 . 2 (∀𝑥(𝑥𝑤 → ∃𝑧𝑦(𝜑𝑦 = 𝑧)) → ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤 ∧ (𝑥𝑤𝜑))))
14 anabs5 850 . . . . . 6 ((𝑥𝑤 ∧ (𝑥𝑤𝜑)) ↔ (𝑥𝑤𝜑))
1514exbii 1772 . . . . 5 (∃𝑥(𝑥𝑤 ∧ (𝑥𝑤𝜑)) ↔ ∃𝑥(𝑥𝑤𝜑))
1615bibi2i 327 . . . 4 ((𝑦𝑧 ↔ ∃𝑥(𝑥𝑤 ∧ (𝑥𝑤𝜑))) ↔ (𝑦𝑧 ↔ ∃𝑥(𝑥𝑤𝜑)))
1716albii 1745 . . 3 (∀𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤 ∧ (𝑥𝑤𝜑))) ↔ ∀𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤𝜑)))
1817exbii 1772 . 2 (∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤 ∧ (𝑥𝑤𝜑))) ↔ ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤𝜑)))
1913, 18sylib 208 1 (∀𝑥(𝑥𝑤 → ∃𝑧𝑦(𝜑𝑦 = 𝑧)) → ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥𝑤𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1479  wex 1702  wnf 1706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-rep 4762
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708
This theorem is referenced by:  zfrepclf  4768  axsep  4771
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